1 Overview

The focus of this document/website is to provide guidance on conducting initial data analysis in a reproducible manner in the context of intended regression analyses.

2 IDA Framework

The IDA framework consists of six steps [Huebner et al 2018, Figure 1], here we assume that metadata (step I) exist in sufficient detail, and that data cleaning (step II) was already performed. Metadata summarize background information about the data to properly conduct IDA steps, and a data cleaning process identifies and corrects technical errors. The data screening (step III) examines data properties to inform decisions about the intended analysis. Initial data reporting (step IV) document insight of the previous steps and can be referred to when interpreting results from the regression modeling. Consequences of these analyses can be that the analysis plan needs to be refined or updated (step V). Finally, reporting of IDA results in research papers (step VI) are necessary to ensure transparency regarding key findings that influence the analysis or interpretation of results. Further details about the elements of IDA are discussed in [TG3 papers].

IDA framework

References

Huebner M, le Cessie S, Schmidt CO, Vach W . A contemporary conceptual framework for initial data analysis. Observational Studies 2018; 4: 171-192. Link

Huebner M, Vach W, le Cessie S, Schmidt C, Lusa L. Hidden Analyses: a review of reporting practice and recommendations for more transparent reporting of initial data analyses. BMC Med Res Meth 2020; 20:61. Link

3 Scope of the regression analyses for the examples

Regression models can be used for a wide range of purposes, for the purpose of these examples the assumptions on the regression analysis set-up in this paper are listed in Table 1. Thus, IDA tasks will be explained in a well-defined, practically relevant setting. Since a key principle is that IDA does not touch the research question no associations between dependent (outcome) and independent (non-outcome) variables are considered.

Table 1: The scope of the regression analyses considered for IDA tasks

Aspects of the research plan	Assumptions in this paper	Reason for the assumption
Dependent (outcome) variable	One dependent variable that can be continuous or binary; exclude time-to-event or longitudinal outcomes	Explain IDA tasks in a well-defined, practically relevant setting
Regression models	Models with linear predictors	Explain IDA tasks in a well-defined, practically relevant setting
Purpose of regression model	Adjust effect of one variable of interest for confounders; quantify the effects of explanatory variables on the outcome	Explain IDA tasks in a well-defined, practically relevant setting
Independent variables	“explanatory” or “confounder” depending on purpose of model; small to moderate number of mixed types; Not high dimensional; no repeated measurements	To demonstrate IDA approaches for a mix of variables likely to be encountered in practice
Statistical analysis plan	Exists, defines the outcome variable, the type of regression model to be used, and a set of independent variables	IDA does not touch the research question, but may lead to an update or refinement of the analysis plan

References:

Vach W. Regression Models as a Tool in Medical Research. Chapman/Hall CRC 2012

Harrell FE. Regression Modeling Strategies. Springer (2nd ed) 2015

Royston P and Sauerbrei W. Multivariable Model Building. Wiley (2008)

[…]

4 Data screening and possible actions

4.1 Univariate distributions

	What to look at	Possible actions: Interpretation	Possible actions: SAP	Possible actions: Presentation
Continuous variables	General skewness	Help in interpreting results	Update SAP	Update intended presentation of results
Continuous variables	General skewness	Wide CI for coefficients	Use variable as log-transformed	Update intended presentation of results
Continuous variables	Outliers	Disproportional impact on results	Winsorize or transform	Model involves winsorization
Continuous variables	Spike at 0	Narrow CI at 0	Use appropriate representation of variable in model	Use 2 (or more) coefficients to distinguish 0 from non-0 continuous part
Categorical variables	Frequencies	Comparisons to default reference probably irrelevant	Change reference category	Contrasts compare to (new) reference category
Categorical variables	Rare categories	Wide CI for coefficients	Collapse/exclude	Fewer categories to present
Categorical variables	One very frequent category	Comparisons irrelevant?	Exclude variable	Variable omitted

4.2 Bivariate distributions

	What to look at	Possible actions: Interpretation	Possible actions: SAP	Possible actions: Presentation
Continuous by continuous	Outliers (from the cloud)	Disproportional impact on results	Winsorize or transform	Model involves winsorization
Continuous by continuous	Correlations	Wide CI for coefficients	Winsorize or transform	Model involves winsorization
Continuous by categorical	Outliers (only visible in bivariate plot)	Wide CI for coefficients
Categorical by categorical	Frequent/rare combinations	Comparison to default reference irrelevant	Change reference category	Contrasts compare to (new) reference category
Categorical by categorical	Frequent/rare combinations	interactions relevant?	Remove interaction from model	Fewer interactions to present

4.3 Missing values

	What to look at	Possible actions: Interpretation	Possible actions: SAP	Possible actions: Presentation
Per variable	Number and proportion	Wide CI for coefficients	Remove variable if many missing values
Pattern	Variables missing independently or together		Omit variables together	Changes model
Pattern	Variables missing dependent on levels of other variables	Systematic missingness? Model still based on representative?	IPW needed?	Weighted analysis
Complete cases	Number and proportion	Few cases left for main CCO analysis	Multiple imputation (or other way of dealing with missing values)?	Result from MI analysis? Or applicability restricted to a subpopulation?

References

Huebner M, le Cessie S, Schmidt CO, Vach W . A contemporary conceptual framework for initial data analysis. Observational Studies 2018; 4: 171-192. Link

Harrell FE. Regression Modeling Strategies. Springer (2nd ed) 2015

[…]

CRASH-2

5 Introduction to CRASH-2

Since a key principle of IDA is not to touch the research questions, before IDA commences the research aim and statistical analysis plan need to be in place. IDA may lead to an update or refinement of the analysis plan. To demonstrate the workflow and content of IDA, we created a hypothetical research aim and corresponding statistical analysis plan, which is described in more detail in the section Crash2_SAP.Rmd.

Hypothetical research aim for IDA is to develop a multivariable model for early death (death within 28 days from injury) using nine independent variables of mixed type (continuous, categorical, semicontinuous) with the primary aim of prediction and a secondary aim of describing the association of each variable with the outcome.

A prediction model was developed and validated based on this data set in “Predicting early death in patients with traumatic bleeding” Perel et al, BMJ 2012, [supplement available at]. The assumed research aim is in line with the prediction model

5.1 CRASH-2 Description

Clinical Randomisation of an Antifibrinolyticin Significant Haemorrhage(CRASH-2) was a large randomised placebo controlled trial among trauma patients with, or at risk of, significant haemorrhage, of the effects of antifibrinolytic treatment on death and transfusion requirement. The study is described at the original trial website. A public version of the data set is found at a repository of public data sets hosted by the Vanderbilt University’s Department of Biostatistics (Prof. Frank Harrell Jr.).

The data set includes 20,207 patients and 44 variables.

Note: In contrast to the analysis described in Perel et al, variables describing the economic region and the treatment allocation are missing in the public version of the data set, and while the data set contains 20,207 patients, the research paper mentions 20,127 patients having been included in the study.

5.2 Crash2 dataset contents

5.2.1 Source dataset

We refer to the source data set as the dataset available online here

Display the source dataset contents. This dataset is in the data-raw folder of the project directory.

Data frame:crash2

20207 observations and 44 variables, maximum # NAs:17121

Name	Labels	Units	Levels	Class	Storage	NAs
entryid	Unique Numbers for Entry Forms			integer	integer	0
source	Method of Transmission of Entry Form to CC		5		integer	0
trandomised	Date of Randomization			Date	double	0
outcomeid	Unique Number From Outcome Database			integer	integer	80
sex			2		integer	1
age					integer	4
injurytime	Hours Since Injury			numeric	double	11
injurytype			3		integer	0
sbp	Systolic Blood Pressure	mmHg		integer	integer	320
rr	Respiratory Rate	/min		integer	integer	191
cc	Central Capillary Refille Time	s		integer	integer	611
hr	Heart Rate	/min		integer	integer	137
gcseye	Glasgow Coma Score Eye Opening			integer	integer	732
gcsmotor	Glasgow Coma Score Motor Response			integer	integer	732
gcsverbal	Glasgow Coma Score Verbal Response			integer	integer	735
gcs	Glasgow Coma Score Total			integer	integer	23
ddeath	Date of Death			Date	double	17121
cause	Main Cause of Death		7		integer	17118
scauseother	Description of Other Cause of Death		227		integer	0
status	Status of Patient at Outcome if Alive		3		integer	3169
ddischarge	Date of discharge, transfer to other hospital or day 28 from randomization			Date	double	3185
condition	Condition of Patient at Outcome if Alive		5		integer	3251
ndaysicu	Number of Days Spent in ICU			numeric	double	182
bheadinj	Significant Head Injury			integer	integer	80
bneuro	Neurosurgery Done			integer	integer	80
bchest	Chest Surgery Done			integer	integer	80
babdomen	Abdominal Surgery Done			integer	integer	80
bpelvis	Pelvis Surgery Done			integer	integer	80
bpe	Pulmonary Embolism			integer	integer	80
bdvt	Deep Vein Thrombosis			integer	integer	80
bstroke	Stroke			integer	integer	80
bbleed	Surgery for Bleeding			integer	integer	80
bmi	Myocardial Infarction			integer	integer	80
bgi	Gastrointestinal Bleeding			integer	integer	80
bloading	Complete Loading Dose of Trial Drug Given			integer	integer	80
bmaint	Complete Maintenance Dose of Trial Drug Given			integer	integer	80
btransf	Blood Products Transfusion			integer	integer	80
ncell	Number of Units of Red Call Products Transfused			numeric	double	9963
nplasma	Number of Units of Fresh Frozen Plasma Transfused			integer	integer	9964
nplatelets	Number of Units of Platelets Transfused			integer	integer	9964
ncryo	Number of Units of Cryoprecipitate Transfused			integer	integer	9964
bvii	Recombinant Factor VIIa Given			integer	integer	374
boxid	Treatment Box Number			integer	integer	0
packnum	Treatment Pack Number			integer	integer	0

Variable	Levels
source	telephone
	telephone entered manually
	electronic CRF by email
	paper CRF enteredd in electronic CRF
	electronic CRF
sex	male
	female
injurytype	blunt
	penetrating
	blunt and penetrating
cause	bleeding
	head injury
	myocardial infarction
	stroke
	pulmonary embolism
	multi organ failure
	other
scauseother
	Acute Hypoxia
	ACUTE LUNG INJURY
	Acute Pulmonary Oedema
	Acute Renal Failure
	ACUTE RESPIRATORY DISTRESS SYNDROME (ARDS)
	acute respiratory failure
	acute respiratory failure+sepsis
	air amboli (embolism)
	Air embolism caused by penetrating lung trauma
	...
status	discharged
	still in hospital
	transferred to other hospital
condition	no symptoms
	minor symptoms
	some restriction in lifestyle but independent
	dependent, but not requiring constant attention
	fully dependent, requiring attention day and night

5.2.2 Updated analysis dataset

Additional meta-data is added to the original source data set. We write this new modified data set back to the data folder after adding additional meta-data for the following variables:

age - add label “Age” and unit “years”.
injury time - add unit “hours”.
total Glasgow coma score - add unit “points”.

At the stage we select the variables of interest to take in to the IDA phase by dropping variables we do not check in IDA.

As a cross check we display the contents again to ensure the additional data is added, and then write back the changes to the data folder in the file “data/a_crash2.rds”.

Input object size: 1221480 bytes; 12 variables 20207 observations New object size: 1223272 bytes; 12 variables 20207 observations Input object size: 1546808 bytes; 14 variables 20207 observations New object size: 1385720 bytes; 14 variables 20207 observations

Data frame:a_crash2

20207 observations and 14 variables, maximum # NAs:17121

Name	Labels	Units	Levels	Class	Storage	NAs
entryid	Unique Numbers for Entry Forms			integer	integer	0
trandomised	Date of Randomization			Date	double	0
ddeath	Date of Death			Date	double	17121
age	Age	years		integer	integer	4
sex	Sex		2		integer	1
sbp	Systolic Blood Pressure	mmHg		integer	integer	320
hr	Heart Rate	/min		integer	integer	137
rr	Respiratory Rate	/min		integer	integer	191
gcs	Glasgow Coma Score Total	points		integer	integer	23
cc	Central Capillary Refille Time	s		integer	integer	611
injurytime	Hours Since Injury	hours		numeric	double	11
injurytype	Injury type		3		integer	0
time2death					integer	17121
earlydeath	Death within 28 days from injury			integer	integer	0

Variable	Levels
sex	male
	female
injurytype	blunt
	penetrating
	blunt and penetrating

5.3 Section session info

## R version 3.6.1 (2019-07-05)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 17763)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=English_United States.1252 
## [2] LC_CTYPE=English_United States.1252   
## [3] LC_MONETARY=English_United States.1252
## [4] LC_NUMERIC=C                          
## [5] LC_TIME=English_United States.1252    
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] Hmisc_4.4-0     Formula_1.2-3   survival_3.2-3  lattice_0.20-40
##  [5] forcats_0.5.0   stringr_1.4.0   dplyr_0.8.5     purrr_0.3.4    
##  [9] readr_1.3.1     tidyr_1.0.2     tibble_3.0.1    ggplot2_3.3.0  
## [13] tidyverse_1.3.0 here_0.1       
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.4.6        lubridate_1.7.4     png_0.1-7          
##  [4] assertthat_0.2.1    rprojroot_1.3-2     digest_0.6.25      
##  [7] R6_2.4.1            cellranger_1.1.0    backports_1.1.7    
## [10] acepack_1.4.1       reprex_0.3.0        evaluate_0.14      
## [13] httr_1.4.1          pillar_1.4.4        rlang_0.4.6        
## [16] readxl_1.3.1        data.table_1.12.8   rstudioapi_0.11    
## [19] rpart_4.1-15        Matrix_1.2-18       checkmate_2.0.0    
## [22] rmarkdown_2.1       splines_3.6.1       foreign_0.8-76     
## [25] htmlwidgets_1.5.1   munsell_0.5.0       broom_0.5.5        
## [28] compiler_3.6.1      modelr_0.1.6        xfun_0.12          
## [31] pkgconfig_2.0.3     base64enc_0.1-3     htmltools_0.4.0    
## [34] nnet_7.3-13         tidyselect_1.1.0    htmlTable_1.13.3   
## [37] gridExtra_2.3       bookdown_0.18       fansi_0.4.1        
## [40] crayon_1.3.4        dbplyr_1.4.2        withr_2.2.0        
## [43] grid_3.6.1          nlme_3.1-145        jsonlite_1.6.1     
## [46] gtable_0.3.0        lifecycle_0.2.0     DBI_1.1.0          
## [49] magrittr_1.5        scales_1.1.1        rmdformats_0.3.7   
## [52] cli_2.0.2           stringi_1.4.6       fs_1.3.2           
## [55] latticeExtra_0.6-29 xml2_1.2.5          ellipsis_0.3.0     
## [58] generics_0.0.2      vctrs_0.3.0         RColorBrewer_1.1-2 
## [61] tools_3.6.1         glue_1.4.1          hms_0.5.3          
## [64] jpeg_0.1-8.1        yaml_2.2.1          colorspace_1.4-1   
## [67] cluster_2.1.0       rvest_0.3.5         knitr_1.28         
## [70] haven_2.2.0

6 Statistical analysis plan

Since a key principle of IDA is not to touch the research questions, before IDA commences the research aim and statistical analysis plan needs to be in place. IDA may lead to an update or refinement of the analysis plan. To demonstrate the workflow and content of IDA, we created a hypothetical research aim and corresponding statistical analysis plan.

Hypothetical research aim for IDA: Develop a multivariable model for early death (death within 28 days from injury) using nine independent variables of mixed type (continuous, categorical, semicontinuous) with the primary aim of prediction and a secondary aim of describing the association of each variable with the outcome.

The assumed analysis aim is in line with the prediction model presented by Perel et al, BMJ 2012, supplement available at.

6.1 Outcome variable

Early death, i.e. in-hospital death within 28 days from injury (binary variable)

6.2 Statistical methods

Logistic regression will be used to model early death by the following independent variables (measured at randomisation) deemed important to predict early death.

Demographic measurements:

Age (age, years)
Sex (sex, male or female)

Physiological measurements:

Systolic blood pressure (sbp, mmHg)
Heart rate (hr, 1/min)
Respiratory rate (rr, 1/min)
Glasgow coma score (gcs, points)
Central capillary refill time (cc, seconds)

Characteristics of injury measurements:

Time since injury (injurytime, hours)
Type of injury (injurytype, ‘blunt’, ‘penetrating’ or ‘blunt and penetrating’)

Restricted cubic splines with 3 degrees of freedom with knots set to default values will be used for continuous variables. As the final prediction model should be parsimonious enough to simplify its application, a backward elimination algorithm with a significance level set at \(\alpha=0.05\) will be applied to remove statistically insignificant effects. Finally, nonlinear representation of each continuous variable will be tested against linear representation at \(\alpha=0.05\). In case of lacking added value of a nonlinear effect, the model will be refitted with a linear effect for that variable.

6.3 Remarks

Regarding type of injury, the original paper describes its treatment in the model as follows: ‘Type of injury had three categories—-penetrating, blunt, or blunt and penetrating—but we analysed it as ’penetrating’ or ‘blunt and penetrating.’ ’ It is not clear from that description what happened to the ‘blunt’ group. (I assume they were collapsed with ‘blunt and penetrating’.) ** we are going to consider the three categories, and then check aout recommendations for the final analysis-MH**
The original paper describes the modeling approach as follows: ‘We used a backward step-wise approach. Firstly, we included all potential prognostic factors and interaction terms that users considered plausible. These interactions included all potential predictors with type of injury, time since injury, and age. We then removed, one at a time, terms for which we found no strong evidence of an association, judged according to the P values (<0.05) from the Wald test.’ This would mean they tested at least 24 interaction terms, each possibly using several degrees of freedom! In the final model, only an interaction of Glasgow coma score and type of injury was included.

6.4 Preparations

The outcome variable, early death (i.e., death within 28 days from injury) must be computed from the time span between date of death and date of randomization using the following logic:

transform ddeath and trandomisation into an interpretable date format and then compute the difference
interpret missing (i.e. NAs) as ‘not died within study period, at least not within 28 days’
if patients died after 28 days, treat as alive

This can be derived using the following code logic:

## NOTE: This is for demostration purposes, this code is not run here. 
## The derivation was executed earlier. 

a_crash2$time2death <-
  as.numeric(as.Date(a_crash2$ddeath) - as.Date(a_crash2$trandomised))

a_crash2$earlydeath[!is.na(a_crash2$time2death)] <-
  (a_crash2$time2death[!is.na(a_crash2$time2death)] <= 28) + 0

# +0 to transform it from TRUE/FALSE to 1/0
# NA in time2death means alive at day 28
a_crash2$earlydeath[is.na(a_crash2$time2death)] <- 0

We also display the marginal distribution of the derived outcome variable.

a_crash2 %>%
  dplyr::select(earlydeath) %>%
  gtsummary::tbl_summary()

Characteristic	N = 20207¹
Death within 28 days from injury	3076 (15%)
¹ Statistics presented: n (%)

The number of deaths computed in the data set coincides with the number reported in Perel et al, BMJ 2012.

6.5 Sources

Data obtained from http://biostat.mc.vanderbilt.edu/wiki/Main/DataSets

To download the data set, click the link to data set

6.5.1 Data dictionary

The data dictionary can be found LINK

6.6 References

CRASH-2 Collaborators. Effects of tranexamic acid on death, vascular occlusive events, and blood transfusion in trauma patients with significant haemorrhage (CRASH-2): a randomised, placebo-controlled trial. Lancet 2010;376:23-32

Perel P, Prieto-Merino D, Shakur H, Clayton T, Lecky F, Bouamra O, Russell R, Faulkner M, Steyerberg EW, Roberts I. Predicting early death in patients with traumatic bleeding: development and validation of prognostic model. BMJ 2012; 345(aug15 1): e5166.

7 Univariate distribution checks

This section reports a series of univariate summary checks of the CRASH-2 dataset.

7.1 Data set overview

Using the Hmisc describe function, we provide an overview of the data set. The descriptive report also provides histograms of continuous variables. For ease of scanning the information, we group the report by measurement type.

7.1.1 Demographic variables

Demographic variables

2 Variables 20207 Observations

age: Age years

n	missing	distinct	Info	Mean	Gmd	.05	.10	.25	.50	.75	.90	.95
20203	4	84	0.999	34.56	15.55	18	19	24	30	43	55	64

lowest : 1 14 15 16 17 , highest: 92 94 95 96 99

sex: Sex

n	missing	distinct
20206	1	2

 Value        male female
 Frequency   16935   3271
 Proportion  0.838  0.162

7.1.2 Physiological measurements

Physiological measurements

5 Variables 20207 Observations

sbp: Systolic Blood Pressure mmHg

n	missing	distinct	Info	Mean	Gmd	.05	.10	.25	.50	.75	.90	.95
19887	320	173	0.989	98.45	27.86	60	70	80	95	110	130	143

lowest : 4 10 12 20 25 , highest: 225 230 234 240 250

hr: Heart Rate /min

n	missing	distinct	Info	Mean	Gmd	.05	.10	.25	.50	.75	.90	.95
20070	137	173	0.996	104.5	23.38	70	80	90	105	120	130	140

lowest : 3 4 5 6 10 , highest: 190 192 198 200 220

rr: Respiratory Rate /min

n	missing	distinct	Info	Mean	Gmd	.05	.10	.25	.50	.75	.90	.95
20016	191	68	0.99	23.06	7.052	14	16	20	22	26	30	35

lowest : 1 2 3 4 5 , highest: 90 91 94 95 96

gcs: Glasgow Coma Score Total points

n	missing	distinct	Info	Mean	Gmd	.05	.10	.25	.50	.75	.90	.95
20184	23	13	0.863	12.47	3.594	4	6	11	15	15	15	15

lowest : 3 4 5 6 7 , highest: 11 12 13 14 15

 Value          3     4     5     6     7     8     9    10    11    12    13    14
 Frequency    784   520   441   584   733   576   504   663   586   951  1356  2140
 Proportion 0.039 0.026 0.022 0.029 0.036 0.029 0.025 0.033 0.029 0.047 0.067 0.106
                 
 Value         15
 Frequency  10346
 Proportion 0.513

cc: Central Capillary Refille Time s

n	missing	distinct	Info	Mean	Gmd	.05	.10	.25	.50	.75	.90	.95
19596	611	20	0.945	3.267	1.67	1	2	2	3	4	5	6

lowest : 1 2 3 4 5 , highest: 17 18 20 30 60

 Value          1     2     3     4     5     6     7     8     9    10    11    12
 Frequency   1510  5328  6020  3367  1805   802   268   271    45   139     3     7
 Proportion 0.077 0.272 0.307 0.172 0.092 0.041 0.014 0.014 0.002 0.007 0.000 0.000
                                                           
 Value         13    15    16    17    18    20    30    60
 Frequency      3    19     3     1     1     2     1     1
 Proportion 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.000

7.1.3 Characteristics of injury

Characteristics of injury

2 Variables 20207 Observations

injurytime: Hours Since Injury hours

n	missing	distinct	Info	Mean	Gmd	.05	.10	.25	.50	.75	.90	.95
20196	11	93	0.972	2.844	2.35	0.5	1.0	1.0	2.0	4.0	6.0	7.0

lowest : 0.10 0.15 0.20 0.25 0.30 , highest: 22.00 45.00 48.00 72.00 96.00

injurytype: Injury type

n	missing	distinct
20207	0	3

 Value                      blunt           penetrating blunt and penetrating
 Frequency                  11189                  6552                  2466
 Proportion                 0.554                 0.324                 0.122

7.2 Categorical variables

We now provide a closer visual examination of the categorical predictors.

7.2.1 Categorical ordinal plots

The Glasgow coma score, an ordinal categorical variable, is also displayed separately.

7.3 Continuous variables

A closer visual examination of continuous predictors.

There is evidence of digit preference. Explore further with targeted summaries. A more detailed univariate summaries for the variables of interest are also provided below.

7.3.1 Age

Distribution of subject age [years]

Five patients under the age of 17, the inclusion criteria for the study, with one patient aged 1.

7.3.2 Blood pressure

Distribution of SBP

7.3.3 Respiratory rate

Distribution of respiratory rate

7.3.4 Heart rate

Distribution of heart rate

7.3.5 Central capillary refill time

Distribution of Central capillary refill time

7.3.6 Hours since injury

Distribution of hours since injury

7.4 Section session info

## R version 3.6.1 (2019-07-05)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 17763)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=English_United States.1252 
## [2] LC_CTYPE=English_United States.1252   
## [3] LC_MONETARY=English_United States.1252
## [4] LC_NUMERIC=C                          
## [5] LC_TIME=English_United States.1252    
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] Hmisc_4.4-0     Formula_1.2-3   survival_3.2-3  lattice_0.20-40
##  [5] forcats_0.5.0   stringr_1.4.0   dplyr_0.8.5     purrr_0.3.4    
##  [9] readr_1.3.1     tidyr_1.0.2     tibble_3.0.1    ggplot2_3.3.0  
## [13] tidyverse_1.3.0 here_0.1       
## 
## loaded via a namespace (and not attached):
##  [1] httr_1.4.1          jsonlite_1.6.1      splines_3.6.1      
##  [4] modelr_0.1.6        assertthat_0.2.1    highr_0.8          
##  [7] latticeExtra_0.6-29 cellranger_1.1.0    yaml_2.2.1         
## [10] pillar_1.4.4        backports_1.1.7     glue_1.4.1         
## [13] digest_0.6.25       RColorBrewer_1.1-2  checkmate_2.0.0    
## [16] rvest_0.3.5         colorspace_1.4-1    htmltools_0.4.0    
## [19] Matrix_1.2-18       pkgconfig_2.0.3     broom_0.5.5        
## [22] haven_2.2.0         bookdown_0.18       patchwork_1.0.0    
## [25] scales_1.1.1        jpeg_0.1-8.1        htmlTable_1.13.3   
## [28] farver_2.0.3        generics_0.0.2      ellipsis_0.3.0     
## [31] withr_2.2.0         nnet_7.3-13         cli_2.0.2          
## [34] magrittr_1.5        crayon_1.3.4        readxl_1.3.1       
## [37] evaluate_0.14       fs_1.3.2            fansi_0.4.1        
## [40] nlme_3.1-145        xml2_1.2.5          foreign_0.8-76     
## [43] tools_3.6.1         data.table_1.12.8   hms_0.5.3          
## [46] lifecycle_0.2.0     munsell_0.5.0       reprex_0.3.0       
## [49] cluster_2.1.0       compiler_3.6.1      rlang_0.4.6        
## [52] grid_3.6.1          rstudioapi_0.11     htmlwidgets_1.5.1  
## [55] base64enc_0.1-3     labeling_0.3        rmarkdown_2.1      
## [58] gtable_0.3.0        DBI_1.1.0           R6_2.4.1           
## [61] gridExtra_2.3       lubridate_1.7.4     knitr_1.28         
## [64] rprojroot_1.3-2     stringi_1.4.6       rmdformats_0.3.7   
## [67] Rcpp_1.0.4.6        vctrs_0.3.0         rpart_4.1-15       
## [70] acepack_1.4.1       png_0.1-7           dbplyr_1.4.2       
## [73] tidyselect_1.1.0    xfun_0.12

8 Multivariate distributions

8.1 Overview

8.1.1 Variable correlation

corrs <- a_crash2 %>%
  dplyr::select(age, sex, sbp, hr, rr ,cc, injurytime, injurytype ) %>%
  filter(complete.cases(.)) %>%
  dplyr::mutate_all(as.numeric)

M <- cor(corrs)
col <- colorRampPalette(c("#BB4444", "#EE9988", "#FFFFFF", "#77AADD", "#4477AA"))
corrplot(M, method = "color", col = col(200),
         type = "upper", order = "hclust", number.cex = .7,
         addCoef.col = "black", # Add coefficient of correlation
         tl.col = "black", tl.srt = 90, # Text label color and rotation
         # hide correlation coefficient on the principal diagonal
         diag = FALSE)

8.1.2 Variable clustering

Variable clustering is used for assessing collinearity, redundancy, and for separating variables into clusters that can be scored as a single variable, thus resulting in data reduction.

Hmisc::varclus( ~ age +  sbp +  hr + rr + cc + gcs + injurytime + injurytype + sex, data = a_crash2)

## Hmisc::varclus(x = ~age + sbp + hr + rr + cc + gcs + injurytime + 
##     injurytype + sex, data = a_crash2)
## 
## 
## Similarity matrix (Spearman rho^2)
## 
##                                  age  sbp   hr   rr   cc  gcs injurytime
## age                             1.00 0.00 0.00 0.00 0.00 0.00       0.01
## sbp                             0.00 1.00 0.11 0.03 0.07 0.01       0.01
## hr                              0.00 0.11 1.00 0.05 0.02 0.02       0.00
## rr                              0.00 0.03 0.05 1.00 0.02 0.00       0.00
## cc                              0.00 0.07 0.02 0.02 1.00 0.02       0.00
## gcs                             0.00 0.01 0.02 0.00 0.02 1.00       0.01
## injurytime                      0.01 0.01 0.00 0.00 0.00 0.01       1.00
## injurytypepenetrating           0.02 0.00 0.01 0.00 0.00 0.06       0.05
## injurytypeblunt and penetrating 0.00 0.01 0.01 0.00 0.00 0.01       0.00
## sexfemale                       0.01 0.00 0.00 0.00 0.00 0.00       0.00
##                                 injurytypepenetrating
## age                                              0.02
## sbp                                              0.00
## hr                                               0.01
## rr                                               0.00
## cc                                               0.00
## gcs                                              0.06
## injurytime                                       0.05
## injurytypepenetrating                            1.00
## injurytypeblunt and penetrating                  0.07
## sexfemale                                        0.02
##                                 injurytypeblunt and penetrating sexfemale
## age                                                        0.00      0.01
## sbp                                                        0.01      0.00
## hr                                                         0.01      0.00
## rr                                                         0.00      0.00
## cc                                                         0.00      0.00
## gcs                                                        0.01      0.00
## injurytime                                                 0.00      0.00
## injurytypepenetrating                                      0.07      0.02
## injurytypeblunt and penetrating                            1.00      0.00
## sexfemale                                                  0.00      1.00
## 
## No. of observations used for each pair:
## 
##                                   age   sbp    hr    rr    cc   gcs injurytime
## age                             20203 19884 20066 20012 19593 20180      20193
## sbp                             19884 19887 19795 19750 19316 19883      19877
## hr                              20066 19795 20070 19943 19482 20066      20059
## rr                              20012 19750 19943 20016 19454 20014      20008
## cc                              19593 19316 19482 19454 19596 19595      19588
## gcs                             20180 19883 20066 20014 19595 20184      20173
## injurytime                      20193 19877 20059 20008 19588 20173      20196
## injurytypepenetrating           20203 19887 20070 20016 19596 20184      20196
## injurytypeblunt and penetrating 20203 19887 20070 20016 19596 20184      20196
## sexfemale                       20202 19886 20069 20015 19595 20183      20195
##                                 injurytypepenetrating
## age                                             20203
## sbp                                             19887
## hr                                              20070
## rr                                              20016
## cc                                              19596
## gcs                                             20184
## injurytime                                      20196
## injurytypepenetrating                           20207
## injurytypeblunt and penetrating                 20207
## sexfemale                                       20206
##                                 injurytypeblunt and penetrating sexfemale
## age                                                       20203     20202
## sbp                                                       19887     19886
## hr                                                        20070     20069
## rr                                                        20016     20015
## cc                                                        19596     19595
## gcs                                                       20184     20183
## injurytime                                                20196     20195
## injurytypepenetrating                                     20207     20206
## injurytypeblunt and penetrating                           20207     20206
## sexfemale                                                 20206     20206
## 
## hclust results (method=complete)
## 
## 
## Call:
## hclust(d = as.dist(1 - x), method = method)
## 
## Cluster method   : complete 
## Number of objects: 10

Plot associations.

plot(Hmisc::varclus( ~ age +  sbp +  hr + rr + cc + gcs + injurytime + injurytype + sex, data = a_crash2))

8.1.3 Variable redundancy

Redundancy analysis of predictor variables.

Hmisc::redun( ~ hr + rr + age + sbp + injurytype + sex  , data = a_crash2)

## 
## Redundancy Analysis
## 
## Hmisc::redun(formula = ~hr + rr + age + sbp + injurytype + sex, 
##     data = a_crash2)
## 
## n: 19689     p: 6    nk: 3 
## 
## Number of NAs:    518 
## Frequencies of Missing Values Due to Each Variable
##         hr         rr        age        sbp injurytype        sex 
##        137        191          4        320          0          1 
## 
## 
## Transformation of target variables forced to be linear
## 
## R-squared cutoff: 0.9    Type: ordinary 
## 
## R^2 with which each variable can be predicted from all other variables:
## 
##         hr         rr        age        sbp injurytype        sex 
##      0.116      0.044      0.052      0.099      0.061      0.035 
## 
## No redundant variables

8.2 Summary reports by sex

8.2.1 Overall

	N	male N=16935	female N=3271
Baseline characteristics by sex.
Age years	20203	23.0 30.0 41.0 33.7 ± 13.6	25.0 35.0 50.0 38.8 ± 16.8
Systolic Blood Pressure mmHg	19887	80.0 95.0 110.0 98.8 ± 25.5	80.0 90.0 110.0 96.7 ± 25.7
Heart Rate /min	20070	90.0 105.0 120.0 104.3 ± 21.2	92.0 106.0 120.0 105.2 ± 21.0
Respiratory Rate /min	20016	20.00 22.00 26.00 23.07 ± 6.77	20.00 22.00 26.00 23.03 ± 6.58
Central Capillary Refille Time s	19596	2.00 3.00 4.00 3.27 ± 1.72	2.00 3.00 4.00 3.23 ± 1.59
Glasgow Coma Score Total points	20184	11.00 15.00 15.00 12.44 ± 3.72	12.00 14.00 15.00 12.62 ± 3.46
Hours Since Injury hours	20196	1.00 2.00 4.00 2.85 ± 2.39	1.00 2.00 4.00 2.84 ± 2.67
Injury type : blunt	20207	0.53 ⁸⁹⁶²/₁₆₉₃₅	0.68 ²²²⁷/₃₂₇₁
penetrating		0.35 ⁵⁹³⁰/₁₆₉₃₅	0.19 ⁶²¹/₃₂₇₁
blunt and penetrating		0.12 ²⁰⁴³/₁₆₉₃₅	0.13 ⁴²³/₃₂₇₁
a b c represent the lower quartile a, the median b, and the upper quartile c for continuous variables. x ± s represents X ± 1 SD. N is the number of non-missing values.

8.2.2 Distribution of age by sex

Distribution of age by sex

8.2.3 Distribution of systolic blood pressure by sex

Distribution of systolic blood pressure by sex

8.2.4 Distribution of heart rate by sex

Distribution of heart rate by sex

8.2.5 Distribution of respiratory rate by sex

Distribution of respiratory rate by sex

8.2.6 Distribution of central capillary refille time by sex

Distribution of centrail capillary refille time by sex

8.2.7 Distribution of hours since injury by sex

Distribution of hours since injury by sex

8.2.8 Distribution of Glasgow comma score by sex

Distribution of Glasgow comma score (point scale) by sex

8.2.9 Distribution of injury type by sex

Distribution of injury type by sex

8.3 Summary reports by age

Categorize age for the purposes of exploring the relationship between age and other baseline variables. This is purely for exploratory purposes only, and not to influence the analysis strategy to pursue the dichotomization of age.

Characteristic	N = 20207¹
age_C
<30	9070 (45%)
30-44	6477 (32%)
45-59	3204 (16%)
60+	1452 (7.2%)
NA	4 (<0.1%)
¹ Statistics presented: n (%)

Report all variables by age category.

	N	<30 N=9070	30-44 N=6477	45-59 N=3204	60+ N=1452
Baseline characteristics by age categories.
Sex : female	20202	0.13 ¹¹⁸³/₉₀₇₀	0.15 ⁹⁵⁹/₆₄₇₆	0.21 ⁶⁵⁹/₃₂₀₄	0.32 ⁴⁶⁹/₁₄₅₂
Systolic Blood Pressure mmHg	19884	80.0 96.0 110.0 98.1 ± 23.8	80.0 90.0 110.0 97.7 ± 25.3	80.0 94.0 112.0 100.1 ± 28.4	80.0 90.0 110.0 100.4 ± 30.2
Heart Rate /min	20066	91.0 106.0 120.0 105.3 ± 21.3	90.0 106.0 120.0 104.7 ± 20.9	90.0 104.0 120.0 103.3 ± 21.0	88.0 100.0 116.0 101.0 ± 21.8
Respiratory Rate /min	20012	20.00 22.00 26.00 22.93 ± 6.74	20.00 22.00 26.00 23.24 ± 6.68	20.00 22.00 26.00 23.11 ± 6.80	20.00 22.00 26.00 23.04 ± 6.89
Central Capillary Refille Time s	19593	2.00 3.00 4.00 3.20 ± 1.77	2.00 3.00 4.00 3.27 ± 1.65	2.00 3.00 4.00 3.34 ± 1.64	2.00 3.00 4.00 3.48 ± 1.56
Glasgow Coma Score Total points	20180	11.00 15.00 15.00 12.64 ± 3.61	11.00 14.50 15.00 12.39 ± 3.72	11.00 14.00 15.00 12.38 ± 3.70	10.00 14.00 15.00 12.00 ± 3.82
Hours Since Injury hours	20193	1.00 2.00 4.00 2.71 ± 2.18	1.00 2.00 4.00 2.83 ± 2.28	1.00 2.50 4.50 3.12 ± 3.17	1.00 3.00 4.50 3.12 ± 2.68
Injury type : blunt	20203	0.50 ⁴⁵⁴⁴/₉₀₇₀	0.53 ³⁴⁶²/₆₄₇₇	0.65 ²⁰⁸¹/₃₂₀₄	0.76 ¹¹⁰¹/₁₄₅₂
penetrating		0.38 ³⁴⁴⁸/₉₀₇₀	0.33 ²¹⁵⁵/₆₄₇₇	0.23 ⁷⁴⁸/₃₂₀₄	0.14 ¹⁹⁹/₁₄₅₂
blunt and penetrating		0.12 ¹⁰⁷⁸/₉₀₇₀	0.13 ⁸⁶⁰/₆₄₇₇	0.12 ³⁷⁵/₃₂₀₄	0.10 ¹⁵²/₁₄₅₂
a b c represent the lower quartile a, the median b, and the upper quartile c for continuous variables. x ± s represents X ± 1 SD. N is the number of non-missing values.

8.3.1 Distribution of systolic blood pressure by age categories

Distribution of systolic blood pressure by gcs

8.3.2 Distribution of heart rate by age categories

Distribution of heart rate by gcs

8.3.3 Distribution of respiratory rate by age categories

Distribution of respiratory rate by gcs

8.3.4 Distribution of central capillary refille time by age categories

Distribution of centrail capillary refille time by gcs

8.3.5 WIP: multivarate scatter plots

a_crash2 %>% dplyr::filter(!is.na(sbp)) %>% tally()

##       n
## 1 19887

a_crash2 %>% dplyr::filter(is.na(sbp)) %>% tally()

##     n
## 1 320

bigN <- a_crash2 %>% dplyr::filter(!is.na(sbp) & !is.na(age)) %>% tally()
n_miss <- a_crash2 %>% dplyr::filter(is.na(sbp) | is.na(age)) %>% tally()

title <-
  paste0("Plot of ", Hmisc::label(a_crash2$age), " and ", Hmisc::label(a_crash2$sbp))

caption <-
  paste0(
    "n = ",
    bigN,
    " subjects displayed.\n",
    n_miss,
    " subjects with a missing value in at least one of the variables."
  )


x_axis <- paste0(Hmisc::label(a_crash2$age), " [", Hmisc::units(a_crash2$age), "]")
y_axis <- paste0(Hmisc::label(a_crash2$sbp), " [", Hmisc::units(a_crash2$sbp), "]")


p1 <- a_crash2 %>%
  dplyr::filter(!is.na(sbp) & !is.na(age)) %>%
  mutate(sbp = as.numeric(sbp), 
         age = as.numeric(age)) %>%
  ggplot(aes(x = sbp, y = age)) +
  ylab(x_axis) +
  xlab(y_axis) +
  labs(
    title = title,
    caption = caption
  ) +
  geom_point(shape = 16, #size = 0.5,
             alpha = 0.5,
             color = "firebrick2") +
  geom_rug() +
  theme_minimal()

p1

8.4 Scatter plots with a third or fourth variable

Scatter plot of age and RR by sex and injury type.

Scatter plot of SBP and RR by sex and injury type.

8.5 Baseline characteristics by Glasgow comma score

	N	3 N=784	4 N=520	5 N=441	6 N=584	7 N=733	8 N=576	9 N=504	10 N=663	11 N=586	12 N=951	13 N=1356	14 N=2140	15 N=10346
Baseline characteristics by Glasgow comma score.
Age years	20203	24.0 32.0 44.0 35.5 ± 14.9	25.0 33.0 44.0 35.5 ± 14.1	24.0 32.0 45.0 35.4 ± 14.7	23.0 31.0 45.0 35.4 ± 15.4	23.0 30.0 42.0 33.9 ± 14.0	24.0 32.0 45.0 35.7 ± 15.0	24.0 32.0 44.0 35.5 ± 14.6	24.0 31.0 42.0 34.4 ± 13.8	24.0 33.0 46.0 36.6 ± 15.6	25.0 32.0 45.0 35.9 ± 14.3	25.0 33.0 45.0 36.4 ± 15.0	24.0 31.0 44.0 35.1 ± 14.7	23.0 30.0 41.0 33.7 ± 13.8
Heart Rate /min	20070	90.0 112.0 128.0 106.9 ± 31.3	95.0 114.0 130.0 110.8 ± 29.2	98.0 110.5 130.0 111.4 ± 25.4	90.0 110.0 123.2 106.2 ± 24.4	95.0 109.0 120.0 107.1 ± 23.0	95.0 110.0 120.0 107.4 ± 24.0	92.0 109.0 120.0 105.5 ± 21.9	96.0 110.0 124.8 108.4 ± 24.0	96.0 110.0 122.0 107.8 ± 20.4	100.0 110.0 122.0 109.3 ± 20.2	96.0 108.0 120.0 106.5 ± 20.1	92.0 105.0 120.0 104.5 ± 19.9	90.0 100.0 115.0 102.0 ± 18.9
Respiratory Rate /min	20016	12.00 20.00 28.00 20.67 ± 10.74	16.00 22.00 28.00 22.22 ± 9.14	18.00 22.00 28.00 22.89 ± 8.69	18.00 21.00 26.00 22.12 ± 7.56	18.00 20.00 26.00 21.97 ± 7.69	18.00 22.00 28.00 23.11 ± 7.73	20.00 24.00 28.00 23.23 ± 6.99	19.00 22.00 28.00 23.05 ± 6.73	20.00 23.00 28.00 23.45 ± 6.37	20.00 24.00 28.00 24.32 ± 6.41	20.00 22.00 27.00 23.45 ± 6.53	20.00 22.00 26.00 23.41 ± 6.09	20.00 22.00 26.00 23.14 ± 6.07
Systolic Blood Pressure mmHg	19887	70.0 85.0 103.0 88.7 ± 33.7	78.0 90.0 116.0 96.5 ± 31.2	80.0 90.0 118.0 99.0 ± 30.7	80.0 100.0 127.0 104.3 ± 32.1	80.0 100.0 130.0 105.4 ± 30.6	80.0 90.0 115.0 99.2 ± 29.4	80.0 96.0 120.0 99.6 ± 28.9	80.0 90.0 110.0 92.6 ± 28.0	80.0 90.0 110.0 94.4 ± 26.4	71.0 90.0 100.0 88.4 ± 24.7	80.0 90.0 110.0 95.9 ± 23.5	80.0 90.0 110.0 96.4 ± 22.8	90.0 100.0 110.0 100.5 ± 23.1
Central Capillary Refille Time s	19596	3.00 4.00 5.00 4.15 ± 2.13	3.00 4.00 5.00 3.84 ± 1.90	2.00 3.00 5.00 3.76 ± 1.91	2.00 3.00 4.00 3.49 ± 1.64	2.00 3.00 4.00 3.28 ± 1.55	2.00 3.00 4.00 3.52 ± 1.69	2.00 3.00 4.00 3.40 ± 3.00	2.00 3.00 4.00 3.37 ± 1.66	2.00 3.00 4.00 3.27 ± 1.51	3.00 3.00 4.00 3.53 ± 1.60	2.00 3.00 4.00 3.40 ± 1.69	2.00 3.00 4.00 3.31 ± 1.73	2.00 3.00 4.00 3.06 ± 1.54
Sex : female	20206	0.14 ¹⁰⁷/₇₈₄	0.13 ⁶⁸/₅₂₀	0.12 ⁵³/₄₄₁	0.16 ⁹²/₅₈₄	0.14 ¹⁰⁰/₇₃₃	0.15 ⁸⁹/₅₇₆	0.15 ⁷⁴/₅₀₄	0.19 ¹²⁴/₆₆₃	0.17 ⁹⁷/₅₈₆	0.21 ¹⁹⁸/₉₅₁	0.20 ²⁷⁰/₁₃₅₆	0.18 ³⁹¹/₂₁₃₉	0.16 ¹⁶⁰⁴/₁₀₃₄₆
Hours Since Injury hours	20196	1.00 2.00 4.00 2.54 ± 1.94	1.00 3.00 5.00 3.26 ± 2.20	1.00 3.00 5.00 3.42 ± 2.19	2.00 3.75 6.00 3.75 ± 2.31	2.00 3.00 5.00 3.62 ± 2.20	1.00 3.00 5.00 3.30 ± 2.17	1.00 3.00 5.00 3.12 ± 2.20	1.00 2.00 4.00 3.03 ± 2.19	1.00 2.50 4.00 3.01 ± 2.05	1.00 2.00 4.00 2.75 ± 2.03	1.00 2.00 4.00 2.79 ± 1.97	1.00 2.00 4.00 2.64 ± 1.99	1.00 2.00 4.00 2.71 ± 2.69
Injury type : blunt	20207	0.62 ⁴⁸³/₇₈₄	0.71 ³⁷¹/₅₂₀	0.73 ³²⁴/₄₄₁	0.76 ⁴⁴³/₅₈₄	0.76 ⁵⁵⁹/₇₃₃	0.69 ³⁹⁹/₅₇₆	0.67 ³³⁸/₅₀₄	0.61 ⁴⁰⁷/₆₆₃	0.64 ³⁷⁷/₅₈₆	0.58 ⁵⁵⁰/₉₅₁	0.60 ⁸¹⁴/₁₃₅₆	0.58 ¹²³⁷/₂₁₄₀	0.47 ⁴⁸⁸⁰/₁₀₃₄₆
penetrating		0.22 ¹⁷⁵/₇₈₄	0.10 ⁵³/₅₂₀	0.09 ⁴¹/₄₄₁	0.10 ⁵⁹/₅₈₄	0.11 ⁷⁷/₇₃₃	0.15 ⁸⁹/₅₇₆	0.17 ⁸⁸/₅₀₄	0.23 ¹⁵¹/₆₆₃	0.21 ¹²³/₅₈₆	0.29 ²⁷²/₉₅₁	0.24 ³²⁶/₁₃₅₆	0.29 ⁶²⁹/₂₁₄₀	0.43 ⁴⁴⁵⁸/₁₀₃₄₆
blunt and penetrating		0.16 ¹²⁶/₇₈₄	0.18 ⁹⁶/₅₂₀	0.17 ⁷⁶/₄₄₁	0.14 ⁸²/₅₈₄	0.13 ⁹⁷/₇₃₃	0.15 ⁸⁸/₅₇₆	0.15 ⁷⁸/₅₀₄	0.16 ¹⁰⁵/₆₆₃	0.15 ⁸⁶/₅₈₆	0.14 ¹²⁹/₉₅₁	0.16 ²¹⁶/₁₃₅₆	0.13 ²⁷⁴/₂₁₄₀	0.10 ¹⁰⁰⁸/₁₀₃₄₆
a b c represent the lower quartile a, the median b, and the upper quartile c for continuous variables. x ± s represents X ± 1 SD. N is the number of non-missing values.

8.5.1 Distribution of age by Glasgow comma score

Distribution of age by gcs

8.5.2 Distribution of systolic blood pressure by Glasgow comma score